Destruction of trust through “commitment” breaking – Rock, Paper, Scissors Tournament

The below New York Times article not only discusses the strategies involved in the often-analyzed game of rock, paper, scissors, but also demonstrates the use of a strategic move during a rock, paper, scissors tournament. In this example, the woman in the tournament is facing her opponent and declares “I’m going to throw rock” – a strategic move called a commitment.  However, as neither player knew one another, the opponent was not sure whether to trust the player making the declaration, wondering if she would really throw rock. “"If she is, she expects me to throw paper, in which case she would throw scissors, in which case I should throw rock. If she really will throw rock, then at worst we would tie."

In this instance, the opponent decided to trust the player and threw rock. Instead, the player threw paper and thus won the first match. However, the two players had to play against one another multiple times. At this point, the first player had destroyed any thought of building trust in the game by not upholding her commitments. The game got progressively worse, and the player who won the first match was eventually eliminated from the game.

This scenario illustrates that short-term gains can definitely be achieved by not upholding strategic moves; however, they do not provide long-term benefits and in many instances may actually create more damage than they create.

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New York Times

September 5, 2004

http://www.nytimes.com/2004/09/05/fashion/05ROCK.html?adxnnl=1&adxnnlx=1260488063-VpGBIduSv/3EusgZWNd0vA&pagewanted=print&position=

 Rock, Paper, Scissors: High Drama in the Tournament Ring

 By JENNIFER 8. LEE

MY opponent and I faced each other across the white lines, separated by an arm's length in the dark, smoky bar. He planted his feet firmly, shoulder-width apart, while I fell into a fighting stance, right foot forward — a natural response from years of tae kwon do. The referee stood between us. The crowd looked on expectantly.

The rules were deceptively simple — rules that people all over the world grasp as young children.

Paper covers rock. Rock crushes scissors. Scissors cut paper.

But like the game Othello, another childhood favorite, Rock Paper Scissors takes a minute to learn and a lifetime to master.

Rock Paper Scissors has gained a cult following in much of the English-speaking world over the last few years. The World Rock Paper Scissors Society, based in Toronto, says that its history dates to London in the mid-1800's and that its membership has grown to 2,300 from 5 since its Web site, www.worldrps.com, first appeared in 1995.

Word of mouth generated by the Web site, and by the world championships that the society has sponsored since 2002, have led to a spread of formal and impromptu tournaments in bars, fraternity houses, homes and high schools. A bar in Chapel Hill, N.C., for example, held a tournament on Aug. 15 that drew 40 competitors. A tournament held for the past two years at the Roshambo Winery in Healdsburg, Calif., has attracted hundreds of spectators and competitors.

"The Official Rock Paper Scissors Strategy Guide," by the brothers Douglas and Graham Walker, the society's directors, will be published next month by Fireside Books, and "Rock Paper Scissors: The Movie," a documentary about the 2003 world championships in Toronto, is to be released in January.

When I decided to compete in a local tournament and started training, some of my friends scoffed at the idea that the game could involve strategy. But this was not the Rock Paper Scissors of the playground, a hurried competition to see which team got the ball first, or even of the fraternity, to see who would go and buy the beer. This was tournament-style Rock Paper Scissors, in which the stakes are high, and expert players do well over time only because of skill and hard work.

There were 128 of us competing for the top three places in the D.C. National Rock Paper Scissors Tournament at DC9, a Washington bar. The first prize was $1,000 and an XM satellite radio, a significant haul — although modest compared to the $31,000 BMW that was awarded at a tournament in Vail, Colo., last April, or to the one million shekels (about $220,000) that a 13-year-old boy won by beating 700 other competitors in an Israeli tournament on Aug. 5.

Advice came to me from all directions. An office-mate offered wisdom gleaned from his days at the frat house: "The key is to throw scissors early and often." Aaron Hoffman, a math graduate student at Brown University, suggested that I counter the risk of overthinking my throws with a seemingly random sequence of numbers. "You could memorize the digits of pi in base 3," he said. "Zero is rock, one is scissors and two is paper." Sure I could.

I called experienced players to ask for tips, and learned about the common tells that can reveal an impending throw. For example, many people will open up paper early. I was told that most people have a go-to throw, reflective of their character, when they are caught off guard. Paper, considered a refined, even passive, throw, is apparently favored by literary types and journalists; I found I was no exception.

I started going up to people at parties and in the office and challenging them to quick matchups. I even attended a training session with the tournament's organizer, Master Roshambollah (also known as Jason Simmons), and some local players.

Over time, my game was getting sharper. I was winning more than I was losing.

But now that I was here in a Washington bar on a Saturday night, surrounded by a crowd of onlookers in their 20's and 30's, I was nervous. Tournament Rock Paper Scissors proceeds a bit like a tennis competition: game, set, match. The first to win two games wins the set, the first to win two sets wins the match. The winner moves on to the next round; the loser, generally, is eliminated. I had never played under the stress of tournament conditions.

Earlier, in my effort to size up and profile the other people in my heat, I had spoken with an opponent, Ryan Taylor. I had been warned of the importance of understanding his personality and level of experience. Mr. Taylor was 23 and had a shaggy 70's-style haircut, and he said he had never played competitively before.

The referee put his hand between us and asked, "Ready?" He reminded us that vertical paper (which resembles a handshake) was a no-no. Many people had fouled on vertical paper over the years. We nodded.

I decided that I would try to psych out Mr. Taylor, using a a technique I had learned from the strategy guide.

I arched my eyebrows, looked him in the eyes, and said slowly, in a flat voice, "I'm going to throw rock."

He seemed momentarily thrown, but then regained his composure.

The referee lifted his hand and we started pumping our fists in sync, the part of the game known as the prime. 

"One. Two. Three. Shoot."

Mr. Taylor was wondering, as he later told me, if I was really going to throw rock: "If she is, she expects me to throw paper, in which case she would throw scissors, in which case I should throw rock. If she really will throw rock, then at worst we would tie."

He threw rock. I threw paper. I won.

What Mr. Taylor didn't realize was that I was playing a defensive rather than offensive game, on the advice of one of the experienced players I had spoken with.

"If you are trying to beat them, you only have one throw" that will work, said Benjamin Stein, a 25-year-old computer programmer in New York. "If you play a defensive game, you have two throws. You can either tie or beat them, and you are successful." Draws are valuable, he explained, because they give you the chance to get more information about your opponent's mindset and strategy. If I could eliminate one of my opponent's three possible next throws, Mr. Stein said, I had a pretty good shot at staying alive.

In this case, I had a strong hunch that Mr. Taylor wouldn't throw scissors, just in case I did actually throw rock. So that left him paper or rock. So I played paper, because that meant I would either tie him (if he played paper) or beat him (if he played rock).

It was logic worthy of "The Princess Bride."

The referee raised his hand to ready us for the next throw.

I tried to divine what Mr. Taylor was planning. I decided he wasn't going to throw rock again after I had just beat him, so scissors was a safe throw for me.

I threw scissors. He threw paper. Game and set were mine. I was only one set from advancing to the next round.

But then I lost the next two throws in rapid succession. Knowing that novices tend to cycle all three throws, I threw a rock, thinking it was time for him to throw scissors. Instead he splayed his hand in flat paper formation. Then his scissors beat my paper. A one-two punch.

We were tied. My heart beat faster. How had I lost a set so quickly?

I gestured for a time-out, to break his momentum and regain my senses. I was disoriented. In the strategy guide, I had read about predetermined three-throw gambits used in competition. I considered a few — Paper Dolls, Fistful of Dollars, the Bureaucrat — before settling on the Avalanche: Rock, Rock, Rock.

I threw my first rock. He went with paper. Ugh. Now it was down to match point.

I breathed deeply. Everything was on the line with this throw.

I threw the second rock. He repeated paper. I was out, eliminated from the tournament.

I saw Mr. Taylor later that night, after he had made it to the top 32. I learned that he had just moved to Washington a few months ago to work at a local theater. (Theater! If I had known that about him, I could have profiled him as a paper guy.)

He offered to buy me a drink, and asked for my number.

I gave it to him. But only because I want a rematch.

 

Ravens Demonstrate Roll-back Equilibrium Path in Hunting Habits

The interesting article below demonstrates the prevalence of game theory players amongst nature. While some could argue whether an animal can be an active game player due to “awareness” issues, assuming that animals hunting qualifies as a game, this bird example provides a fascinating application of sequential gaming and hints at animals’ ability to determine best responses. This particular instance is important as game theorists modeled this best response prior to it actually ever being observed in nature, pointing to game theory as an evolutionary predictor.

As the article explains, “teenage” ravens are at a serious disadvantage when hunting for food compared with adult ravens. Older, more experienced couples claim animal carcasses for themselves. If a younger bird attempts to share or claim a portion of the carcass, the older birds fight with it until it flies away. Thus, some of the younger birds have begun to hunt in packs with “friends”. This sets up a sequential game whereas the young raven can choose to try to steal the carcasses from the older birds alone or with a pack; in both situations, the older birds have the choice to stay and fight or to fly away.

While the game theorists mentioned in the article do not share their game theory model, I have attempted to construct one as demonstrated in “Hunting Habits of Jedi “Teenage” Ravens.” This model illustrates the two choices in the game and shows how the adult ravens best response to a single teenage raven is to stay and fight but to flee if a pack of ravens approach. Using roll-back analysis, the roll-back eqilibrium for the teenage raven is then to hunt with friends.

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http://www.physorg.com/news162200932.html

Raven teenager gangs play by game theory

May 22, 2009

 

Game theory models predicted that young ravens gain the greatest advantage from hunting in a pack. Now the young birds have figured this out for themselves, and form gangs to oust older raven pairs from tasty carcasses.

 

Young ravens have transformed their hunt for carrion into a joint venture. Instead of searching on their own, some young ravens form bands of as many as 30 birds.

“This behaviour in young ravens has not been documented before, and came as a surprise to us. But the fun was that our game theory model predicted this behaviour as an alternative strategy for young ravens”, says Jonathan Wright, a biology professor at the Norwegian University of Science and Technology (NTNU).

Wright has been studying the behaviour of young ravens in Wales with his colleague Sasha Dall, of the University of Exeter in Cornwall.

Information Central

Ravens feed on the carcasses of large animals. In rural areas in Wales, this basically means sheep. Adult raven pairs defend the cadavers that are found in their territories, while young ravens gather in large, communal roosts - like youth hang-outs - and cooperate extensively in their hunt for food.

Wright and Dall studied one of the world’s largest roosts for these birds, with roughly 2000 ravens. Their study reinforces the theory that the roosts act as information centres where the birds share information about where to find food.

Chase off older birds                      

“It is most common for young ravens to look for food as individuals. Their efforts can then be combined to cover a large area. In the evening they return to the roost. The birds who have found a dead sheep to tell the other birds about share this information with a sort of flight demonstration of how far away the food is”, says Wright.

The birds recruit six or seven mates, and fly off the next day to the carcass, chase off the older bird pair, and help themselves to the food. The day afterwards, six or seven young ravens return to the carcass, and the process continues until the food is eaten up.

The territorial, nesting pair will defend their areas. But there is little they can do against a hungry mob of young birds.

“The adult pair that has been chased away stays in the background until the young birds are done for the day and have returned to the roost”, Wright says.

Hanging out with their mates

The researchers set out sheep cadavers at different distances from the communal roost, and put small plastic beads into the flesh, with different colours for each carcass.

“A crappy job, it was nice to have students to do this”, says Wright, smiling.

The ravens ate the meat with plastic beads, and when they returned to the roost in the evening, the beads were regurgitated. The different colours of the plastic beads documented quite clearly that ravens that had eaten from the same cadaver also stayed at the same roost.

“While we observed this, we also found these gangs of birds that behaved in a completely different way. Instead of searching one by one, they hunted for food together, in a party of perhaps 30 birds. These gangs hunted near the roost. Because they covered a smaller area, they didn’t have the same need to split up to hunt for food. As a bunch they could easily chase away a nesting couple from a cadaver, and they did not need to wait a day for the first bird to recruit other young ravens from the roost.”

Young couples

The researchers found clusters of plastic beads of the same colour a little away from the main area where the rest of the young ravens were gathered. The gang had thus roosted a little away from the other birds.

“I think that the gang consists of slightly older young birds that have already formed couples. They are sort-of ‘engaged’, but not yet ‘married’, because they have not found a free nesting territory. But this is only a theory,” says Wright.

Predicted by theory

Dall and Wright used a game theory mathematical model to predict how the young ravens would behave.

“Our model had foreseen both an individual exploration strategy for food far away from the roost and a gang strategy for food near the roost. We predicted that the gang strategy would be optimal where there is usable food near the roosting area. But we first thought that part of the model was wrong, because no one has observed these kinds of raven gangs before,” says Wright.

Game theory uses abstract mathematical models to analyse different players’ choices between alternative strategies. It is commonly applied in economic theory, but also in subjects such as political science, sociology and anthropology.

And now in biology.

“We have long used game theory in evolutionary biology research, when we study behavioural development in an ecological context. But the raven study is a very good example of how game theory can help understand how behaviour and social structures may evolve under changing conditions. And it's especially funny that it could predict a behaviour that had not yet been observed in reality”, says Wright.

 

The Office – Conflict Resolution: Zero sum games vs. Non-zero sum games

The popular television series, The Office, aired an episode entitled “Conflict Resolution” which deals with a clash between two co-workers over a poster that one worker, Angela, has hung in the accounting desk area; the poster features babies playing musical instruments. Angela’s co-worker, Oscar, on the other hand, finds the poster offensive and disturbing and wants the poster taken down. (See the 2-min clip at the end of this blog post for highlights from the episode or visit: http://www.nbc.com/The_Office/video/clips/conflict-resolution/116196/)

This episode illustrates an example of a zero sum game as if the poster is taken down, Angela loses, and Oscar wins. Pam, the receptionist, suggests a solution to alleviating the conflict. Angela and Oscar can split the time with the poster, with the poster staying up half the week and taken down for the other half. This, however, is still considered zero-sum as Angela wins half the time and Oscar loses while the second half of the week Oscar wins and Angela loses for a total sum of “0”.

Michael, Angela and Oscar’s boss, attempts to solve the dilemma by converting it into a non-zero sum game, thus creating a “win-win” situation. Michael suggests that the poster be made into a t-shirt that Oscar will wear so that he can never see the design and Angela can always see it. This would entail creative cooperation between the two parties. However, both parties agree that this is a ridiculous suggestion and insist on returning to the original parameters of the game.

Not only does the episode demonstrate the difference in zero-sum games vs. non-zero sum games, but it also reveals the potential of a mediator to advance a game. While these two coworkers did not come to an agreement, Michael was able to look at the situation from a different perspective than the two players to provide alternative solutions that they might not be able to see.

Is Collective Social Action through Weak Facebook Ties Feasible

Social media is currently a rapidly expanding area, the uses for which are quickly moving beyond the strictest sense of the word “social” with companies, products, and causes all interacting in the social media sphere. A Facebook application launched last year attempted to motivate broad groups of people by allowing users to develop an idea for an ultimatum to enact while setting a “tipping point” and tracking pledged support. Once that critical mass of support has been pledged, the collective group will follow-through on the designated action. For instance, the article below provides the example that if a million people pledge support for a call for Walmart to provide healthcare benefits, all million people will boycott the company if the demand is not met.

While the desire to empower others and to work for social change is honorable, the effectiveness of this new tool is questionable when seen through the eyes of game theory, with two major concerns coming to the surface. The first point of contention lies in the use of ultimatums. Ultimatums are in essence a form of threats, a common game theory technique for “changing” the rules of the game in one’s favor. However, to have proper effect, threats must be credible. The Facebook application as a behavioral motivator has not yet earned a reputation for upholding collectively pledged actions; similarly, as any one person can become the instigator of an ultimatum in this application, that person’s individual qualifications as a leader are also not established. Therefore, the targeted organizations may not respond to the threat; thus, requiring the group to perform the designated action. 

Inspiring collective action amongst as diverse a group of people as are on Facebook also has serious flaws through the lens of game theory. In the Walmart example, employees of the store or friends of employees may sign up; however, when the million supporter level is reached for the boycott, there is no enforcement mechanism to ensure compliance. Therefore, individuals are faced with the choice of contributing or assuming that the other 999,999 people will continue and the importance of that individual’s level of action is minimal. The following game table I developed explores this idea:

		Everyone Else
		Contribute	Don’t Contribute
Me	Contribute	5,5	-5, 0
	Don’t Contribute	6,5	0,0

 

In this case, “me” has a greater payout if “everyone else” does all the work and “me” reaps the benefit. “Me has a dominant strategy of not contributing; and “everyone else” has a dominant strategy of contributing anyway providing “me” with the incentive not to act. However, if every individual signed up for the ultimatum uses this game table view, everyone will adopt a policy of not acting because they assume everyone else will and carrying out the ultimatum will fail.

Therefore, while the intentions of this application are honorable, the credibility of threats issued by Facebook users is questionable. At the time of writing this post, I could no longer find this application available on Facebook, but have no details as to what led to its removal.

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http://www.wired.com/underwire/2008/02/group-activism/

New Facebook App Lets Groups Issue Ultimatums

By Jenna Wortham  February 27, 2008  |  2:33 pm  |  Categories: Uncategorized

A new Facebook app called Ultimatums lets users float an idea, set an arbitrary "tipping point" and track group members’ commitment to their common goal. Once a critical mass of people has pledged support, a predetermined action — like a boycott of Ford Motor Company or plans to only use nickels to purchase  public transit fares — is taken.

The app, launched Tuesday by Andrew Mason, founder of consumer activism site The Point, could help mobilize the group potential of Facebook’s 66 million users.

"People think their efforts will be wasted, that their contribution doesn’t make a difference," said Mason, who dropped out of graduate school to pursue his online, grass-roots push for social change. "But here, each campaign is a success in progress."

From rallying support for war-torn countries like Darfur to organizing student protests and flash-mob-style mass purchases of vegetables, Facebook groups have been used to instigate all kinds of social action. But unlike Facebook application Causes, which lets users donate money to a favorite nonprofit, Ultimatums goes beyond uniform models.

Instead of abstract ideas (like Facebook group Make Poverty History),
Ultimatums lets users set concrete goals. The end result is a straightforward app designed to transform a vibrant online community into a vehicle for specific social change. For example, one group calls for Wal-Mart to provide health care benefits for its employees. If a million users sign up, all million pledge to boycott the company if the demand isn’t met.

Each ultimatum is an online petition created by users that requires a tipping point of participants to induce action. Participants are committed to act only when enough other individuals agree to the same, so support for each cause is bound by the group commitment. Mason says he hopes this will eliminate a common downside of collective action — the feeling that one person can’t effect change.

Since launching the Facebook component, Mason estimates 100 new ultimatums have been created, calling for everything from the humane treatment of animals by fast food corporations to DRM-free music and an end to late fees for video rentals.

The service is still in it’s nascent stages; most of the petitions have yet to break 100. And most successes have been on a smaller scale:
raising money for small sports tournaments or a Nicaraguan farm collective. But Mason says he remains hopeful that with each ultimatum, the potential for large-scale change looms closer.

"If we can get just enough people involved, we can force change," he said.

Soap Operas Make a Business Out of Cooperation in Prisoner’s Dilemmas

Since the age of 7 when my summer babysitter would provide me with the option of taking a nap or quietly sitting with her as she watched her afternoon soap operas, the drama and excitement of the storylines combined with the familiarity of characters across a continuous narrative have inspired an amazing sense of loyalty to General Hospital even twenty years later. While watching a new “wrinkle” unfold over the last several months, I realized that soap operas like General Hospital are not only filled with game theory, but that the “famous” prisoner’s dilemma in many ways is an essential building block for the television genre.

For instance, several months ago, “Man A” (pictured above) uncovered a secret that would potentially destroy his wife “Woman A,” (also pictured) threatening not only her health but also the life of their unborn child. At the same time, “Woman B” discovered that same secret; in her case, the surfacing of the secret could result in the potential mob killing of her current boyfriend “Man B.” Through a series of events, Man A and Woman B realized that they both were keeping the same secret. At this point, they were faced with a prisoner’s dilemma. In both cases, a tremendous amount of damage would be created if they both revealed the secret to their loved ones; however, if their loved ones learned of the secret from the other knowing party rather than their trusted loved one, the individuals would be in an even worse position as the initial forecasted damage could happen with the addition of greater harm to themselves. (For example, if Woman A learns the secret from Woman B rather than her own husband, not only will her and her child’s lives be in jeopardy but she may never trust her husband again.) If, however, they both choose not to tell, then they must live with the guilt, which also may threaten to destroy their relationships. While the Nash equilibrium is that both should tell, the two secret holders realize that they can avoid a great deal of the danger and risk if they both continue to keep the secret. Therefore, in this particular instance, Man A and Woman B attempted to cooperate together to cover-up the life-changing secret rather than playing their dominant strategies. (See the game table below)

		Man A
		Tell	Don’t Tell
Woman B	Tell	-10, -10	-3, -15
	Don’t Tell	-15, -3	-5, -5

While cooperation in real-life is extremely difficult, “attempted cooperation” is an essential element to the longevity of soap operas as this is the element that allows shows like General Hospital to extend their storylines. The cooperation enables the show to tryout new wrinkles while building more suspense around the “secret-reveal moment.” For instance, for this example the loved ones of Man A and Woman B began to suspect that they were having an affair as they were spending a great deal of time with one another while covering up the secret. It also reduces the workload of the writing staff as they would have to produce exponentially more storylines should every secret be revealed at the time of discovery. Finally, it also allows for more “opportune” reveal moments in story lines adding the essential element of drama for which soap operas are known.

Looking briefly at the cooperation break-down factor, soap operas do also illustrate the difficulty in maintaining cooperation between parties. In the months prior to the secret being revealed, audiences see both Woman B and Man A considering breaking down and telling their loved ones the truth. The situation also demonstrates the debilitating effect that the entry of a third player can have on cooperation. In this case, a third party (Woman A’s son who woke up from a coma and remembered a bedside confession regarding the secret) entered the game and made the secret public. Woman B then let slip to the astonished town that Man A had known for months, resulting in Woman A declaring her intention to leave her husband (right before she goes into labor and is kidnapped by the mob.)

While some may debate the educational component of soap opera content, this avid supporter is impressed by the real-world game-theory connections that can be found if one only looks beyond the surface.

For a preview of the next strategic game in store now that Woman A has been kidnapped, watch the following promo: 

Efficiency Attempt in Organ Allocation Create “Fairness” Questions

“And yet, there are ways to work the system to one’s advantage. Waiting times for a liver vary in different parts of the country, and people who can afford to travel are free to go to a city or state with the shortest wait and bide their time until they have reached the top of the list, a donor dies and an organ becomes available. Indeed, some patients rent apartments or stay in hotels near a hospital and wait for the phone to ring. It may not seem fair, but it is not illegal.”

 Steve Job’s liver transplant several months ago raises questions regarding the division of the scarce resource of human organs to individuals. The current distribution system is run by UNOS and is based on two overarching principles of allocation:  medical utility and justice, with the principle of medical utility dictating that organs should be allocated to “create the greatest good for the greatest number of patients” while distributive justice dictates that organs  “should be allocated to patients in the greatest need for a transplant or who would benefit most” (such as children).  Thus, a point system similar to the example at the top of this post of how kidneys are allocated is used to determine priority recipients:

While the existing system uses this point system to achieve a level of “equitability” in the allocation of organs, the system’s attempts to achieve efficiency have created a loop hole that wealthy patients such as Jobs can use to their own advantage. Because of the costs of transporting organs as well as the risk of damaging the organs, the United States is divided into different regions for organ allocation which attempts to create the greatest number of successes (or pie) given the current set of resources. However, because of differing proportions of patients to organs in the specific regions, wait times for organs vary tremendously from just a few months to over a year or more. No rules exist to prevent individuals from registering in more than one region for an organ; therefore, those individuals with the resources to quickly travel between regions at a moment’s notice have a distinct advantage in the system. This advantage for the rich then fosters “envy” among others on the waitlist.

While this is certainly a complicated issue, UNOS may want to look at refining the rules surrounding registering at multiple centers so as to ensure a more equitable division between economic classes.

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http://www.nytimes.com/2009/06/23/business/23liver.html

New York Times:  A Transplant That Is Raising Many Questions

By DENISE GRADY and BARRY MEIER

Published: June 22, 2009

Reports that Apple’s chief executive, Steven P. Jobs, traveled to Tennessee for a liver transplant about two months ago raise many questions — not just about his prognosis, but also about the system for allocating scarce organs to the many people who need them.

Whenever someone rich and famous receives a transplant, suspicions inevitably arise about whether that person managed to jump to the head of the waiting list and take an organ that might have saved the life of somebody just as desperate but less glamorous. The dark theories are a holdover from the case of Mickey Mantle, who waited all of one day for a liver in 1995, and then died from liver cancer anyway, just two months later.

In Mr. Jobs's case, doctors say there was no need, and little opportunity, to cheat the system. Under current procedures, any transplant center ranks potential liver recipients on its waiting list, with the highest rankings based on how sick the patients are and how long they have been that sick. Jumping ahead of a sicker patient is not allowed.

And yet, there are ways to work the system to one’s advantage. Waiting times for a liver vary in different parts of the country, and people who can afford to travel are free to go to a city or state with the shortest wait and bide their time until they have reached the top of the list, a donor dies and an organ becomes available. Indeed, some patients rent apartments or stay in hotels near a hospital and wait for the phone to ring. It may not seem fair, but it is not illegal.

It is even conceivable that someone could go to the time and expense of registering for the waiting lists of several transplant centers around the country.

“If you had access to a jet and had six hours to get anywhere in the country, you’d have a wide choice of programs,” said Dr. Michael Porayko, the medical director of liver transplants at Vanderbilt University, one of the Tennessee centers that has said it did not treat Mr. Jobs.

Mr. Jobs’s transplant, but not the location of the hospital or the details of his treatment, has been confirmed by people briefed on the matter by current and former Apple board members. Mr. Jobs has declined to comment.

Some doctors say that for Mr. Jobs, a transplant may have been a wise move medically — though others say the evidence is inconclusive.

Experts said that the most likely reason for Mr. Jobs to need a transplant would be that pancreatic cancer, for which he has disclosed having surgery in 2004, had spread, or metastasized, to his liver.

“If you were to postulate why he did it, I think the most likely reason would be that he had liver metastasis,” said Dr. Richard M. Goldberg, an expert on pancreatic cancer at the University of North Carolina, Chapel Hill, who is not involved in Mr. Jobs’s treatment.

Though other, noncancerous types of liver disease could also have led to a transplant, experts say cancer is the most likely explanation.

The liver is the most common site for the spread of pancreatic cancer, especially the rare kind that Mr. Jobs had, known as a neuroendocrine tumor, Dr. Goldberg said. That type of tumor tends to be slow-growing and far more treatable than the more common type of pancreatic cancer, which can be fatal within months.

When neuroendocrine tumors do metastasize, Dr. Goldberg said, they often spread only to the liver, rather than all over the body, and a transplant may be recommended.

Often, though, when tumors spread to the liver, surgeons can treat them by removing just part of the liver. The fact that Mr. Jobs needed a transplant suggests that he might have had diffuse disease throughout his liver, something that does not bode well, Dr. Goldberg said.

“The prognosis for somebody with metastatic liver disease is not nearly as good as for somebody who has disease confined to the pancreas,” Dr. Goldberg said.

“I think this confirms the speculation that there was more going on than had been previously acknowledged,” he said, “but it still doesn’t really tell us where things are likely to go from here.”

Dr. Goldberg said no large-scale, controlled studies had been done on the benefits of transplants for this type of cancer.

“It’s controversial whether transplant is a helpful approach,” he said.

Dr. Lewis Teperman, the director of transplant surgery and vice chairman of surgery at the Langone Medical Center of New York University, says that transplants are frequently done for people with certain types of liver cancer. About half the center’s liver transplants involve cancerous organs, he said, though not usually metastatic cancers.

According to one national study, more than half the patients receiving transplants for cancerous livers were still alive after five years.

A transplant would be reasonable for treating metastases of the kind of pancreatic cancer Mr. Jobs had, Dr. Teperman said, adding that if Mr. Jobs’s liver had had been “full of the tumor,” the transplant would prolong his life.

“But I can’t tell you how much, because I don’t know the extent of the tumor,” Dr. Teperman said. Like Dr. Goldberg, he was not involved in Mr. Jobs’s care.

Unfortunately, Dr. Teperman said, the medicines needed to prevent rejection of the transplant could allow the tumor to regrow.

“There may be some cancer cells scattered around, and they tend to come back to the new liver,” he said.

Some people can avoid the transplant waiting list entirely by receiving a transplant from a live donor, usually a friend or relative who volunteers to give up part of a liver. But Dr. Teperman said he would not recommend that kind of surgery for people with extensive liver tumors, because they might also need a vein that could not be removed from a live donor. They are better off waiting for a liver from a cadaver, he said.

The nation’s organ donor system is managed by the United Network for Organ Sharing, or UNOS, a nonprofit group in Richmond, Va., that operates under contract with the federal government.

When an organ like a liver becomes available, the UNOS database is searched for patients who need it most and have registered at the transplant center where the organ is available, said Anne Paschke, a spokeswoman for the group.

To qualify for an organ, a patient must have first been examined and approved for a transplant by a doctor at that center. Patients often register with a transplant center close to where they live or are directed to register with a specific center by their insurance company, Ms. Paschke said.

But there is nothing, apart from cost, to prevent a patient from registering with multiple centers throughout the country, Ms. Paschke said. A person with generous insurance coverage, or who themselves can afford to pay for the medical work-ups and the transplant, would have an advantage.

There is a trove of publicly available data that can help patients calculate waiting times and get other information about transplant centers nationwide.

For example, in Tennessee, the state in which Mr. Jobs reportedly had his transplant, the median waiting time varies from 3.8 months at Methodist Hospital in Memphis to 17.2 months at Vanderbilt University Medical Center in Nashville, according to a transplant registry operated by Arbor Research Collaborative for Health and the University of Michigan. The median waiting time nationally is 12.3 months, according to that registry.

Ms. Paschke said that the only way a patient could acquire a donated organ outside the system would be if a donor directed that an organ be given to a specific person.

A scoring system used by UNOS, known as a MELD score, determines where a patient ranks on a transplant waiting list. The higher the score, which runs from 6 to 40, the sicker a patient is and the higher the ranking. Any ties are broken by who has had that score the longest.

But patients with high MELD scores on the East or the West Coast will wait longer than those with the same scores in the Midwest or elsewhere, because high demand means fewer organs are available on the coasts, Dr. Porayko said.